The brain uses its own previous activity to adapt to an ever-changing environment. This history dependent adaptation takes place at all scales of organization of the nervous system. The objective of this project is to develop a common theoretical formalism to be applied to multiple history dependent phenomena, from the biochemical reactions that underlie synaptic plasticity, to the emergent patterns in complex neural networks. At the core of this formalism is the recognition that most models of neuronal activity are based on the classical reaction-diffusion equation. Aim 1 will prove that a generalization of this equation, the fractional order reaction-diffusion equation, provides all the natural mathematical tools to incorporate history dependence into neuronal function analysis. The fractional order exponent of the fractional derivative is a parameter that captures the emergent interactions of multiple elements that cause the history dependent process. Systems of equations will be developed for specific cases of activation of membrane conductances, the membrane voltage, and firing rate activity. The objective of Aim 2 is to demonstrate the significance of using this formalism in four applications across neurobiological scales, from synaptic plasticity to sensory processing. Application 1 will focus in understanding history dependence in the biochemical reactions that underlie synaptic long-term depression in cerebellar Purkinje cells as a function of the intracellular structure of dendritic spines. Application 2 will establish a collaboration with a group at the Allen Brain Institute to build fractional order models that can replicate the variability observed in their Cell Types Database of mouse cortical neurons. Application 3 will be a collaboration with a group at the Max Plank Institute for Dynamics and Self-Organization to test the hypothesis that history-dependent neuronal elements give rise to robust reverberating networks. Finally, Application 4 is a collaboration with a group at McGill University. In this case, experiments in awake weakly electric fish will be conducted to determine the cellular and network biophysical substrates that implement optimal coding of sensory input due to fractional differentiation. Overall, this project will provide a unified theoretical framework and develop applications to study, analyze, and design experiments of history dependent neuronal activity across scales.